Standing waves on a string (review) n=1,2,3... Different boundary conditions: Both ends fixed (see above) Both ends free (similar to both ends fixed ) One end fixed and on end free (next slide)

One end fixed and on end free n=1 n=3 Standing waves on a string (review)

Standing waves in tubes (longitudinal) Waves in tubes (pipes) can be described in terms of: displacement vibrations of the fluid pressure variations in the fluid A pressure node is a displacement antinode and vice versa Open and both ends closed pipes n=1 n=2 n=3 Closed (both ends): displacement pressure Open: pressure displacement

One end open pipes (stopped pipes) n=1 n=3 displacementpressure Example: Standing sound waves are produced in a pipe that is 0.6 m long. For the first overtone, determine the locations along the pipe (measured from the left end) of the displacement nodes. The pipe is closed at the left end and open at the right end. Nodes at 0.0 m and 0.4 m

Sound Acoustic waves in the range of frequencies: 20Hz -20,000Hz Sound waves: can travel in any solid, liquid or gas travel faster in a medium that is more dense in liquids and gases sound waves are longitudinal ONLY! longitudinal and transversal sound waves could propagate in in solids Sound in air is a longitudinal wave that contains regions of low and high pressure Vibrating tuning fork These pressure variations are usually small – a “loud” sound changes the pressure by 2.0x10 -5 atm Pressure sensor

Speed of Sound Speed of waves in strings (review): Speed of sound waves: B is bulk modulus, defined as is density Speed of sound waves in a gas: is ratio of heat capacities, is the equilibrium pressure of the gas

Example: Speed of Sound in Air

Speed of Sound in Some Common Substances 1. Air (20 o C) 344 Substance Speed (m/s) 3. Water 1, Human tissue 1, Aluminum 5, Iron and steel 5, Lead 1, Helium 1,006

Pressure variations

Example: Displacement Wave Amplitude of Sound in Air

We classify sounds according to their waveforms: piano note Properties of Sound - A “pure tone” is a sound with a sinusoidal waveform. - This is a sound with a single frequency; produced by tuning fork, etc. - A “complex tone” is a sound that repeats itself but is not sinusoidal Most sounds are like this! A note from a musical instrument will be mostly sinusoidal, but have a character all its own that is specific to the instrument. “Noise” is sound with a complex waveform that does not repeat No definite wavelength or frequency

We use three qualities to characterize how we perceive sound: Perception of Sound 3. The tone quality of a sound is how we distinguish sounds of the same pitch and loudness (how we perceive the qualities of the waveform) 1. The pitch of a sound is how “high” or “low” we perceive a sound to be (directly related to how we perceive frequency); combinations of notes that are “pleasing” to the ear have frequencies that are related by a simple whole- number ratio (Pythagoras) 2. The loudness of a sound is how we perceive the amplitude of the sound wave 1. Pitch. 2. Loudness. 3. Tone quality. The unit of sound loudness is the decibel (dB) Quiet: 30 dB; Moderate: 50 dB; Noisy: 70 dB; Very loud: 90 dB; Problems: 120 dB

Anything that causes pressure vibrations creates sound! Musical Instruments Stringed musical instruments produce sound by vibrating a wire or string A plucked string on a guitar produces a different sound than a violin The tension in the string is used to adjust the wave speed and the frequency Wind instruments produce sound by pressure waves in a tube The sound reflects partly at the open end Valves change the effective length of the tube

Complex Waves To produce a pure fundamental tone on a guitar string: Pull the template away fast. Then the string will vibrate in its fundamental mode. Pluck a guitar string in the middle: Fourier Theorem: The initial shape is a superposition of the sinusoidal shapes for the fundamental mode and for higher harmonics. The sound produced is the fundamental frequency plus higher harmonics. Different musical instruments sound different even when playing the same tone partly because the harmonic contents are different.

Fourier Analysis